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39
A Graduated Assignment Algorithm for Graph Matching
, 1996
"... A graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise. By combining graduated nonconvexity, twoway (assignment) constraints, and sparsity, large improvements in accuracy and speed are achieved. Its low order computational comp ..."
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Cited by 285 (15 self)
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A graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise. By combining graduated nonconvexity, twoway (assignment) constraints, and sparsity, large improvements in accuracy and speed are achieved. Its low order computational complexity [O(lm), where l and m are the number of links in the two graphs] and robustness in the presence of noise offer advantages over traditional combinatorial approaches. The algorithm, not restricted to any special class of graph, is applied to subgraph isomorphism, weighted graph matching, and attributed relational graph matching. To illustrate the performance of the algorithm, attributed relational graphs derived from objects are matched. Then, results from twentyfive thousand experiments conducted on 100 node random graphs of varying types (graphs with only zeroone links, weighted graphs, and graphs with node attributes and multiple link types) are reported. No comparable results have...
A DoubleLoop Algorithm to Minimize the Bethe and Kikuchi Free Energies
 NEURAL COMPUTATION
, 2001
"... Recent work (Yedidia, Freeman, Weiss [22]) has shown that stable points of belief propagation (BP) algorithms [12] for graphs with loops correspond to extrema of the Bethe free energy [3]. These BP algorithms have been used to obtain good solutions to problems for which alternative algorithms fail t ..."
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Cited by 108 (4 self)
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Recent work (Yedidia, Freeman, Weiss [22]) has shown that stable points of belief propagation (BP) algorithms [12] for graphs with loops correspond to extrema of the Bethe free energy [3]. These BP algorithms have been used to obtain good solutions to problems for which alternative algorithms fail to work [4], [5], [10] [11]. In this paper we rst obtain the dual energy of the Bethe free energy which throws light on the BP algorithm. Next we introduce a discrete iterative algorithm which we prove is guaranteed to converge to a minimum of the Bethe free energy. We call this the doubleloop algorithm because it contains an inner and an outer loop. It extends a class of mean eld theory algorithms developed by [7],[8] and, in particular, [13]. Moreover, the doubleloop algorithm is formally very similar to BP which may help understand when BP converges. Finally, we extend all our results to the Kikuchi approximation which includes the Bethe free energy as a special case [3]. (Yedidia et al [22] showed that a \generalized belief propagation" algorithm also has its xed points at extrema of the Kikuchi free energy). We are able both to obtain a dual formulation for Kikuchi but also obtain a doubleloop discrete iterative algorithm that is guaranteed to converge to a minimum of the Kikuchi free energy. It is anticipated that these doubleloop algorithms will be useful for solving optimization problems in computer vision and other applications.
The Softassign Procrustes Matching Algorithm
 Information Processing in Medical Imaging
, 1997
"... . The problem of matching shapes parameterized as a set of points is frequently encountered in medical imaging tasks. When the pointsets are derived from landmarks, there is usually no problem of determining the correspondences or homologies between the two sets of landmarks. However, when the poin ..."
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Cited by 60 (4 self)
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. The problem of matching shapes parameterized as a set of points is frequently encountered in medical imaging tasks. When the pointsets are derived from landmarks, there is usually no problem of determining the correspondences or homologies between the two sets of landmarks. However, when the point sets are automatically derived from images, the difficult problem of establishing correspondence and rejecting nonhomologies as outliers remains. The Procrustes method is a wellknown method of shape comparison and can always be pressed into service when homologies between pointsets are known in advance. This paper presents a powerful extension of the Procrustes method to pointsets of differing point counts with correspondences unknown. The result is the softassign Procrustes matching algorithm which iteratively establishes correspondence, rejects nonhomologies as outliers, determines the Procrustes rescaling and the spatial mapping between the pointsets. 1 Introduction One of the mos...
Data clustering using a model granular magnet
 Neural Computation
, 1997
"... We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a d ..."
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Cited by 57 (2 self)
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We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a decreasing function of the distance between the neighbors. This magnetic system exhibits three phases. At very low temperatures, it is completely ordered; all spins are aligned. At very high temperatures, the system does not exhibit any ordering, and in an intermediate regime, clusters of relatively strongly coupled spins become ordered, whereas different clusters remain uncorrelated. This intermediate phase is identified by a jump in the order parameters. The spinspin correlation function is used to partition the spins and the corresponding data points into clusters. We demonstrate on three synthetic and three real data sets how the method works. Detailed comparison to the performance of other techniques clearly indicates the relative success of our method. 1
Replicator Equations, Maximal Cliques, and Graph Isomorphism
, 1999
"... We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to fo ..."
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Cited by 53 (11 self)
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We present a new energyminimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear onetoone correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the socalled replicator equations—a class of straightforward continuous and discretetime dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate meanfield annealing heuristics.
A Robust Point Matching Algorithm for Autoradiograph Alignment
, 1997
"... We present a novel method for the geometric alignment of autoradiographs of the brain. The method is based on finding the spatial mapping and the onetoone correspondences (or homologies) between point features extracted from the images and rejecting nonhomologies as outliers. In this way, we atte ..."
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Cited by 38 (12 self)
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We present a novel method for the geometric alignment of autoradiographs of the brain. The method is based on finding the spatial mapping and the onetoone correspondences (or homologies) between point features extracted from the images and rejecting nonhomologies as outliers. In this way, we attempt to account for the local natural and artifactual differences between the autoradiograph slices. We have executed the resulting automated algorithm on a set of left prefrontal cortex autoradiograph slices, specifically demonstrated its ability to perform point outlier rejection, validated it using synthetically generated spatial mappings and provided a visual comparison against the well known iterated closest point (ICP) algorithm. Visualization of a stack of aligned left prefrontal cortex autoradiograph slices is also provided.
SelfOrganizing Maps: Generalizations and New Optimization Techniques
 Neurocomputing
, 1998
"... We offer three algorithms for the generation of topographic mappings to the practitioner of unsupervised data analysis. The algorithms are each based on the minimization of a cost function which is performed using an EM algorithm and deterministic annealing. The soft topographic vector quantization ..."
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Cited by 30 (1 self)
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We offer three algorithms for the generation of topographic mappings to the practitioner of unsupervised data analysis. The algorithms are each based on the minimization of a cost function which is performed using an EM algorithm and deterministic annealing. The soft topographic vector quantization algorithm (STVQ)  like the original SelfOrganizing Map (SOM)  provides a tool for the creation of selforganizing maps of Euclidean data. Its optimization scheme, however, offers an alternative to the heuristic stepwise shrinking of the neighborhood width in the SOM and makes it possible to use a fixed neighborhood function solely to encode desired neighborhood relations between nodes. The kernelbased soft topographic mapping (STMK) is a generalization of STVQ and introduces new distance measures in data space based on kernel functions. Using the new distance measures corresponds to performing the STVQ in a highdimensional feature space, which is related to data space by a nonlinear ma...
Neural Optimization
 The Handbook of Brain Research and Neural Networks. Bradford Books/The
, 1998
"... Introduction Many combinatorial optimization problems require a more or less exhaustive search to achieve exact solutions, with the computational effort growing exponentially or worse with system size. Various kinds of heuristic methods are therefore often used to find reasonably good solutions. Th ..."
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Cited by 29 (2 self)
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Introduction Many combinatorial optimization problems require a more or less exhaustive search to achieve exact solutions, with the computational effort growing exponentially or worse with system size. Various kinds of heuristic methods are therefore often used to find reasonably good solutions. The artificial neural network (ANN) approach falls within this category. In contrast to most other methods, the ANN approach does not fully or partly explore the discrete statespace. Rather, it "feels" its way in a fuzzy manner through an interpolating, continuous space towards good solutions, and allows for a probabilistic interpretation. Key elements in this approach are the meanfield (MF) approximation (Hopfield and Tank, 1985; Peterson and S¨oderberg, 1989), annealing, and for many problems the Potts formulation (Peterson and S¨oderberg, 1989). Recently, also propagator methods have proven most valuable for handling
A Lagrangian Relaxation Network for Graph Matching
 IEEE Trans. Neural Networks
, 1996
"... A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing ..."
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Cited by 26 (7 self)
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A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions." Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetrybreaking selfamplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and selfamplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variable...